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Russian Journal of Physical Chemistry B

On a trajectory method for evaluating the transition probability in a quantum system

Valery V Smirnov

Published Mar 1, 2012• 5 citations
Publisher

Abstract

A version of calculation of the transition probability in a quantum system based on the representation of the transition probabilities in the form of a path integral and its evaluation by the saddle-point method for part of the variables. Some problems associated with the analytic continuation of the integrand are discussed. The proposed approach is compared to related methods widely used in chemical physics.

DOI
10.1134/s1990793112020236

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A version of calculation of the transition probability in a quantum system based on the representation of the transition probabilities in the form of a path integral and its evaluation by the saddle-point method for part of the variables. Some problems associated with the analytic continuation of the integrand are discussed. The proposed approach is compared to related methods widely used in chemical physics.
In present work are considered the scheme of evaluation the transition probability in quantum system. It is based on path integral representation of transition probability amplitude and its evaluation by means of a saddle point method, applied to the part of integration variables. The whole integration process is reduced to initial value problem solutions of Hamilton equations with a random initial phase point. The scheme is related to the semiclassical initial value representation approaches using great number of trajectories. In contrast to them from total set of generated phase paths only one path for each initial coordinate value is selected in Monte Karlo process.
The variant of transition probability evaluation in a quantum system, based on transition probability representation by a path integral using a density matrix formalism, is described. To estimate an integral's value, a set of variables is selected and the Monte Carlo method is used for integration over these variables. The stationary phase method is applied to integration over the remaining variables. In the case of the final state with a diagonal density matrix, this scheme leads to a simple expression for transition probability.
An introduction is given to the methodology of the path-integral formulation of quantum Monte Carlo simulation. This allows us to reformulate a d −dimensional partition function of a quantum mechanical system into the form of a d +1−dimensional path-integral written in classical mechanical variables. In addition, the quantum mechanical partition function is shown to be related by a process of analytical continuation to the time-development operator of the quantum mechanical system. The evaluation of the path-integral representation is discussed for the Maxwell-Boltzmann ideal gas, a chain of atoms, and liquid 4 He . The properties of liquid Helium I and Helium II are discussed in terms of the two-fluid model, the Ginzburg theory, and recent quantum Monte Carlo Simulation studies. The properties of liquid 3 He are discussed in the context of recent analytical p-wave pairing theories which are compared to the theory of s-wave pairing in the theory of superconducting electron.
As an alternative formulation of quantum mechanics, path integral is based on the notion of transition amplitude which gives the wave function of a quantum system at a time t f by acting on the wave function at an earlier time ti.We show that for a general quadratic form for the Lagrangian of the system, transition amplitude has the form f (t f -ti)e i h S class., where S class. is the classical action.We then present an algebraic method to evaluate the function f (t f -ti) without refereing to the path integral calculations.We examine the presented method to the cases of free particle and harmonic oscillator and obtain their propagators.
In this chapter we show that it is possible to describe quantum mechanics by using path integrals. This idea was depeloped by Richard Feynman on the basis of an earlier intuition of Paul Dirac. We discuss the path integral approach starting with the problem of a single quantum particle in the presence of an external potential. We introduce the concept of the saddle point and the stationary phase approximation, which is solved by using Fresnel integrals. We show that Gaussian integrals are crucial in the imaginary time formulation of the path integral, which is the appropriate framework for studying quantum systems at thermal equilibrium. Finally, we analyze the quantum tunneling problem, deriving an elegant formula for the tunneling probability based on the simple, but meaningful, saddle point approximation.